Hilbert space renormalization for the many-electron problem

TL;DR

This paper introduces Hilbert space renormalization and the HS-MPS wavefunction ansatz as new tools for describing many-electron correlations, offering a flexible and efficient approach compared to traditional methods.

Contribution

It develops the concept of Hilbert space renormalization and the HS-MPS framework, expanding the theoretical and computational toolkit for many-electron problems.

Findings

HS-MPS provides low-rank tensor approximations to CI spaces.

Truncating the virtual dimension yields size-extensive wavefunctions.

Numerical comparisons show HS-MPS's advantages over traditional methods.

Abstract

Renormalization is a powerful concept in the many-body problem. Inspired by the highly successful density matrix renormalization group (DMRG) algorithm, and the quantum chemical graphical representation of configuration space, we introduce a new theoretical tool: Hilbert space renormalization, to describe many-electron correlations. While in DMRG, the many-body states in nested Fock subspaces are successively renormalized, in Hilbert space renormalization, many-body states in nested Hilbert subspaces undergo renormalization. This provides a new way to classify and combine configurations. The underlying wavefunction ansatz, namely the Hilbert space matrix product state (HS-MPS), has a very rich and flexible mathematical structure. It provides low-rank tensor approximations to any configuration interaction (CI) space through restricting either the 'physical indices' or the coupling rules in the HS-MPS. Alternatively, simply truncating the 'virtual dimension' of the HS-MPS leads to a family of size-extensive wave function ansaetze that can be used efficiently in variational calculations. We make formal and numerical comparisons between the HS-MPS, the traditional Fock-space MPS used in DMRG, and traditional CI approximations. The analysis and results shed light on fundamental aspects of the efficient representation of many-electron wavefunctions through the renormalization of many-body states.